Differential Scanning Calorimeters measure the heat flow to a sample as the sample temperature is varied in a controlled manner. There are two basic types of DSCs, heat flux and power compensation. Brief descriptions of the two types of DSC are included below. A detailed description of the construction and theory of DSCs is disclosed in xe2x80x9cDifferential Scanning Calorimetry an Introduction for Practitionersxe2x80x9d, G. Hxc3x6hne, W. Hemminger and H.-J. Flammersheim (Springer-Verlag, 1996).
Heat flux DSCs include a sensor to measure heat flow to a sample to be analyzed. The sensor has a sample position and a reference position. The sensor is installed in an oven whose temperature is varied dynamically according to a desired temperature program. As the oven is heated or cooled, the temperature difference between the sample and reference positions of the sensor is measured. This temperature difference is assumed to be proportional to the heat flow to the sample.
Power compensation DSCs include a sample and a reference holder installed in a constant temperature enclosure. Each of the holders has a heater and a temperature sensor. The average of the sample and reference holder temperatures is used to control temperature, which follows the desired temperature program. In addition, differential power proportional to the temperature difference between the holders is added to the average power to the sample holder and subtracted from the average power to the reference holder in an effort to reduce the temperature difference between sample and reference holders to zero. The differential power is assumed to be proportional to the sample heat flow and is obtained by measuring the temperature difference between the sample and reference holder. In commercial power compensation DSCs, the difference between sample and reference temperature is generally not zero because a proportional controller is used to control the differential power.
Modulated DSC (MDSC) is a technique for measuring heat flow in a Differential Scanning Calorimeter in which the DSC is subjected to periodic temperature oscillations superimposed on the constant heating rate segments of conventional DSC. MDSC is described in U.S. Pat. No. 5,224,775, which is incorporated herein by reference. In a preferred embodiment, the resultant oscillating sample heat flow signal is separated into reversing and nonreversing components. The current MDSC heat flow measurement requires an increasingly large heat capacity calibration factor to obtain correct results as the modulation period decreases. This frequency dependence limits applicability to relatively long periods. The present invention is an MDSC method that eliminates, or greatly reduces the frequency dependence. This reduces uncertainty in heat capacity and in the reversing and nonreversing heat flow signals. It also increases productivity by allowing the user to use shorter periods and, consequently, higher underlying heating rates.
To carry out a measurement on a DSC, the sample to be analyzed is loaded into a pan and placed on the sample position of the DSC. An inert reference material may be loaded into a pan and placed on the reference position of the DSC, although usually the reference pan is empty. The temperature program for conventional DSCs typically includes combinations of linear temperature ramps and constant temperature segments. Modulated DSC uses a temperature program in which periodic temperature oscillations are superposed on linear ramps and isothermal segments. The experimental result is the sample heat flow versus temperature or time. The heat flow signal is the result of heat flow to or from the sample due to its specific heat and as a result of transitions occurring in the sample.
During the dynamic portion of the DSC experiment, a temperature difference is created between the sample and reference positions of the DSC. In heat flux DSCs, the temperature difference results from the combination of three differential heat flows: the difference between the sample and reference heat flow, the difference between sample and reference sensor heat flow and the difference between sample and reference pan heat flow. In power compensation DSCs, the temperature difference results from the combination of three differential heat flows plus the differential power supplied to the sample holders: the difference between the sample and reference heat flow, the difference between sample and reference holder heat flow and the difference between sample and reference pan heat flow. The heat flow difference between the sample and reference consists of heat flow due to the heat capacity difference between the sample and reference, the heat flow of a transition, or the difference in heating rate that occurs during an MDSC experiment. The heat flow difference between the sample and reference sections of the DSC is the result of thermal resistance and capacitance imbalances in the sensor or between the holders and the difference in heating rate that occurs between the sample and reference sections of the DSC during a transition or during an MDSC experiment. Similarly, the heat flow difference between the sample and reference pans is the result of mass differences between the pans and the difference in heating rate that occurs during a sample transition or during a MDSC experiment.
In conventional heat flux DSCs the sensor imbalance and pan imbalance are assumed to be insignificant and the differences in heating rates are ignored. In conventional power compensation DSCs the holder imbalance and pan imbalance are assumed to be insignificant and the differences in heating rates are ignored. When the balance assumptions are satisfied and the sample heating rate is the same as the programmed heating rate, the temperature difference is proportional to the sample heat flow and the differential temperature gives an accurate measure of the sample heat flow. The sample heat flow is only proportional to the measured temperature difference between sample and reference when the heating rate of the sample and reference are identical, the sensor is perfectly symmetrical, and the pan masses are identical. Proportionality of sample heat flow to temperature difference for a balanced sensor and pans occurs only during portions of the experiment when the instrument is operating at a constant heating rate, the sample is changing temperature at the same rate as the instrument and there are no transitions occurring in the sample. During Modulated DSC experiments, the heating rates of the sample and reference are generally not the same and the difference between measured sample and reference temperatures is not proportional to the sample heat flow.
Thus, the sample heat flow from a conventional DSC is not the actual sample heat flow, but includes the effects of imbalances and differences in heating rates; in other words the DSC sample heat flow measurement is smeared. For many DSC experiments, the smeared sample heat flow is a sufficiently accurate result. For example, when the desired experimental result is the total energy of the transition, such as the heat of fusion of a melt, the total peak area is integrated over a suitable baseline and the result from a conventional DSC is sufficiently accurate. If however, partial integration of the peak area is required (for example, in the study of reaction kinetics), the smeared sample heat flow of conventional DSC cannot be used. Another example of when the conventional DSC result is inadequate is when two or more transitions in a sample occur within a small temperature interval. In that case, the transitions may be poorly separated in prior art DSCs because of the smearing effects. The improvement in resolution of the present invention greatly improves the separation of closely spaced transitions. In any case, the heat flow signal from prior art DSCs does not accurately portray the sample heat flow during a transition.
During a transition, the heat flow to the sample increases or decreases from the pre-transition value depending upon whether the transition is exothermic or endothermic and whether the DSC is being heated or cooled. The change in sample heat flow causes the heating rate of the sample to be different from that of the DSC and as a consequence, the sample pan and sensor heating rates become different from the programmed heating rate.
U.S. patent applications Ser. Nos. 09/533,949 and 09/643,870, incorporated by reference above, disclose a heat flux DSC that uses a four term heat flow equation to account for sensor imbalances and differences in heating rate between the sample and reference sections of the sensor. The four term DSC heat flow equation derived in the ""949 application is:   q  =            Δ      ⁢              xe2x80x83            ⁢                        T          0                ·                  (                                                    R                r                            -                              R                s                                                                    R                r                            ·                              R                s                                              )                      -                  Δ        ⁢                  xe2x80x83                ⁢        T                    R        r              +                  (                              C            r                    -                      C            s                          )            ·                        ⅆ                      T            s                                    ⅆ          τ                      -                  C        r            ·                                    ⅆ            Δ                    ⁢                      xe2x80x83                    ⁢          T                          ⅆ          τ                    
The first term accounts for the effect of the difference between the sensor sample thermal resistance and the sensor reference thermal resistance. The second term is the conventional DSC heat flow. The third term accounts for the effect of the difference between the sensor sample thermal capacitance and the sensor reference thermal capacitance. The fourth term accounts for the effect of the difference between the heating rates of the sample and reference sides of the DSC.
U.S. patent application Ser.No. 09/643,869, incorporated by reference above, discloses a power compensation DSC that uses a five term heat flow equation to account for sample and reference holder imbalances and differences in heating rate between the sample and reference holders. The five term power compensation DSC heat flow equation derived in the ""869 application is:   q  =            Δ      ⁢              xe2x80x83            ⁢      p        +          Δ      ⁢              xe2x80x83            ⁢                        T          0                ·                  (                                                    R                r                            -                              R                s                                                                    R                r                            ·                              R                s                                              )                      -                  Δ        ⁢                  xe2x80x83                ⁢        T                    R        r              +                  (                              C            r                    -                      C            s                          )            ·                        ⅆ                      T            s                                    ⅆ          τ                      -                  C        r            ·                                    ⅆ            Δ                    ⁢                      xe2x80x83                    ⁢          T                          ⅆ          τ                    
The first term is the difference in power supplied to the sample position versus the power supplied to the reference position. The second term accounts for differences between the thermal resistances of the sample and reference holders. The third term accounts for the heat flow that results from the difference in temperature between the sample and reference. The fourth term is the heat flow resulting from imbalances in thermal capacitance between the sample and reference holders. The fifth term reflects heat flow resulting from differences in heating rate between the sample and reference holders.
Heat flow results from that invention show improved dynamic response and hence improved resolution along with improvements in the DSC baseline heat flow. However, the heat flow signal obtained from the practice of that invention still includes the effects of the sample pans.
Modulated Differential Scanning Calorimetry
Modulated Differential Scanning Calorimetry (MDSC) can be used with both heat flux and power compensation DSCs. In MDSC, the temperature program of the DSC cell consists of periodic temperature fluctuations superimposed on a conventional program comprising constant heating rate segments. The measured heat flow is periodic, having the same period as the temperature program while the amplitude and phase angle of the heat flow signal change in response to the sample heat flow. In the preferred embodiment, sample heat flow is comprised of reversing and nonreversing components. The reversing heat flow component is the result of heat storage due to the specific heat capacity of the sample, while the remainder of the total heat flow is the nonreversing heat flow component. Separation of the total heat flow signal into reversing and nonreversing components consists of determining the reversing heat flow, which is subtracted from the total heat flow, leaving the nonreversing heat flow.
The heat capacity of a body is the product of mass and specific heat capacity and is defined by:       q    =          C      ·                        ⅆ          T                          ⅆ          τ                      ,
where C is the heat capacity, q is rate of heat flow to or from the body, T is the temperature of the body and xcfx84 is time. For MDSC, the applied temperature modulation is generally sinusoidal, so the sample temperature is also sinusoidal:   T  =                              T          _                ·                  sin          ⁢                      (                                          ω                ·                τ                            -              φ                        )                              ⁢              xe2x80x83            ⁢      ω        =                  2        ·        π            P      
The bar over the T indicates temperature amplitude; P is the period of the modulation and xcfx86 is the phase angle between the applied and the resultant temperature modulations. Differentiate temperature and substitute into the heat capacity equation:
q=Cxc2x7{overscore (T)}xc2x7xcfx89xc2x7cos(xcfx89xc2x7xcfx84xe2x88x92xcfx86)
Obviously, the heat flow must also be a cosine for the equality to hold:
{overscore (q)}xc2x7cos(xcfx89xc2x7xcfx84xe2x88x92xcfx86)=Cxc2x7{overscore (T)}xc2x7xcfx89xc2x7cos(xcfx89xc2x7xcfx84xe2x88x92xcfx86)
The bar over q indicates heat flow amplitude. Solve for the heat capacity:   C  =            q      _              ω      ·              T        _            
This equation is used to find the sample heat capacity, which is used to calculate the reversing heat flow in MDSC. The MDSC deconvolution algorithm as described in U.S. Pat. No. 5,224,775 determines the temperature and heat flow amplitudes. The reversing heat flow is the heat capacity multiplied by the underlying heating rate:       q    rev    =      C    ·          ⟨                        ⅆ          T                          ⅆ          τ                    ⟩      
The ( ) brackets indicate averaging over an interval comprising an integer multiple of periods, with evaluation at the central point of the interval.
To obtain the correct value of the sample heat capacity from the measurement, a heat capacity calibration factor is applied to the measured heat capacity:       C    p    =            K      c        ·                  q        _                    ω        ·                  T          _                    
The calibration factor is strongly dependent upon period, as shown in FIG. 1 for isothermal MDSC experiments with 0.05 K and 0.2 K temperature amplitudes, and as explained in A. Boller, Y. Jin, B. Wunderlich xe2x80x9cHeat Capacity Measurement by Modulated DSC at Constant Temperaturexe2x80x9d, Journal of Thermal Analysis, Vol. 42 (1994) 307-330. Such large calibration factors suggest that the heat capacity equation used in MDSC is not a good model for sample heat flow. As is known to one of ordinary skill in this field, the heat capacity equation is not limited to sinusoidal oscillations. It may be applied to any periodic oscillation.
Nonreversing heat flow is just the total averaged heat flow minus the reversing component:
qnon=qtotxe2x88x92qrev
It is clear that the key to obtaining accurate separation of the reversing and nonreversing heat flow is accurate calculation of the sample heat capacity.
The heat capacity equation is conventionally used in MDSC to find the reversing heat flow. It is a differential heat flow, i.e., it is the difference in heat flow between the sample and an inert reference:
q=qsxe2x88x92qr
Consider the specific heat capacity equation for two bodies, the sample and its pan and a reference (if used) and its pan:       q    s    =                              C          sm                ·                              ⅆ                          T              sm                                            ⅆ            τ                              ⁢              xe2x80x83            ⁢              q        r              =                  C        rm            ·                        ⅆ                      T            rm                                    ⅆ          τ                    
With a sinusoidal temperature modulation applied, the differential heat flow as measured by DSC is:
q=Csmxc2x7{overscore (Tsm)}xc2x7cos(xcfx89xc2x7xcfx84xe2x88x92xcfx86sm)xe2x88x92Crmxc2x7{overscore (Trm)}xc2x7cos(xcfx89xc2x7xcfx84xe2x88x92xcfx86rm)
This equation cannot be solved for sample heat capacity resulting in a simple equation using the heat flow and temperature amplitudes, because the sample and reference heat flows are not in phase, and because the sample and reference temperature amplitudes differ.
The total sample heat capacity consists of the sample heat capacity plus the pan heat capacity:
Csm=Css+Csp
Css is the sample heat capacity and Csp is the sample pan heat capacity.
Similarly, the total reference heat capacity consists of the reference heat capacity plus the pan heat capacity:
Crm=Crs+Crp
Crs is the reference heat capacity and Crp is the reference pan heat capacity.
If the period of the modulation is relatively large and the sample mass is small, the phase and amplitude difference between the sample and reference temperatures become small and the differential heat flow equation becomes:
q=(Csmxe2x88x92Crm)xc2x7{overscore (Tsm)}xc2x7cos(xcfx89xc2x7xcfx84xe2x88x92xcfx86sm )
This equation can be solved to find the difference between sample and reference heat capacity in the same manner as the heat flow equation for a single body. If the reference pan is empty, Crs=0 and:
Csmxe2x88x92Crm=Css+Cspxe2x88x92Crp
If the sample and reference pans have the same mass:
Css=Csmxe2x88x92Crm
And the measured sample heat capacity is close to the correct value; i.e. the heat flow calibration factor Kc shown in FIG. 1 is very nearly one. This explains why the heat capacity calibration factor is close to one for long modulation periods.
The present invention can be applied to either heat flux or power compensation DSCs that can independently measure the sample and reference heat flows, and that account for differences in heating rate between the sample and reference pans and the difference in heating rate between the sample and reference (if a reference is used). FIGS. 1 and 2 are schematic diagrams of thermal network models for heat flux DSCs and power compensation DSCs, respectively.
Heat Flux DSCs
As applied to heat flux DSCs, the present invention measures the differential heat flow to the sample based upon a single absolute temperature measurement and two differential temperature measurements. Differential scanning calorimeters of the present invention have substantially improved resolution over conventional instruments, with an empty-cell heat flow that is much closer to zero than that obtained in conventional instruments.
Temperature Measurements
In the present invention, the differential heat flow to the sample with respect to the reference is calculated from measurements of the absolute temperature of the base of the sensor, the differential temperature between the sample position and the base of the sensor, and the differential temperature between the sample and reference positions. The differential temperatures are measured using a sample temperature detector (e.g., a sample area temperature detector), a reference temperature detector (e.g., a reference area temperature detector) and a base temperature detector.
The base temperature detector (which measures the temperature of the base of the sensor near its connection to the oven) is used to control the oven temperature. The sample temperature is measured by measuring the difference between the sample temperature and the base temperature, and subtracting that difference from the base temperature to obtain the sample temperature, i.e., the sample temperature is obtained from Ts=T0xe2x88x92xcex94T0. By making a single absolute temperature measurement T0, and a differential temperature measurement between the base and sample positions, any relative errors in absolute temperature measurements due to differences in temperature sensors are eliminated. Also, this structure minimizes the drift of sample temperature during isothermal segments. The heat flow signal that results from this structure has improved baseline performance and improved dynamic response. Additionally, because the heat flow signal is greater during a transition, the calorimeter has greater sensitivity.
The sensor constructed according to the present invention exhibits improved independence between the sample and reference positions. For example, whereas in a prior art heat flux instrument, a deviation in the temperature of the sample position of 13.4% was observed when a sample of indium is placed on the reference position and heated through the melt, in an exemplary experiment with the present invention that deviation is only about 1.4%, i.e., the present invention exhibits an improvement by about an order of magnitude over the prior art instruments. Thus sensors constructed according to the present invention are xe2x80x9ceffectively independent,xe2x80x9d because they exhibit a temperature deviation at the sample position when an indium sample is placed on the reference position of less than about 1.5% of the temperature deviation at the sample position when an indium sample is placed on the sample position.
Calibration
In a first preferred embodiment of the present invention, the differential scanning calorimeter of the present invention is calibrated by running two separate experiments. These experiments determine the four sensor thermal parameters, Cs (the sensor sample thermal capacitance), Cr (the sensor reference thermal capacitance), Rs (the sensor sample thermal resistance) and Rr (the sensor reference thermal resistance) experimentally, and thus calibrate the heat flow sensor.
The first experiment is performed with an empty DSC cell. The DSC cell is first held at an isothermal temperature that is below the temperature range of the calibration, for a time segment sufficient to ensure complete equilibration of the sensor. The DSC cell is then heated at a constant heating rate to a temperature above the temperature range of the calibration, and then held at that temperature for another isothermal segment, for a time segment sufficient to ensure equilibration of the sensor at that temperature. This first experiment is used to calculate the sample and reference time constants as a function of temperature over the calibrated temperature range.
The heat flow balance equation for the sample side of the sensor is:       q    s    =                              T          0                -                  T          s                            R        s              -                  C        s            ·                        ⅆ                      T            s                                    ⅆ          τ                    
where xcfx84 represents time, qs is the heat flow to the sample and the sample pan, Rs is the sensor sample thermal resistance, and Cs is the sensor sample thermal capacitance. Similarly, the heat balance equation on the reference side of the sensor is:       q    r    =                              T          0                -                  T          r                            R        r              -                  C        r            ·                        ⅆ                      T            r                                    ⅆ          τ                    
where qr is the heat flow to the reference and the reference pan, Rr is the sensor reference thermal resistance, and Cr is the sensor reference thermal capacitance.
The heat flow to the sample and the heat flow to the reference should be zero (since the DSC cell is empty). Accordingly, if qs and qr are set equal to zero in the heat balance equations for the sample and reference sides of the sensor, the time constants for the sample and reference are given by:             τ      s        =                            C          s                ⁢                  R          s                    =                        Δ          ⁢                      xe2x80x83                    ⁢                      T            0                                    (                                    ⅆ                              T                s                                                    ⅆ              τ                                )                      and            τ      r        =                            C          r                ⁢                  R          r                    =                                    Δ            ⁢                          xe2x80x83                        ⁢                          T              0                                +                      Δ            ⁢                          xe2x80x83                        ⁢            T                                                              ⅆ                              T                s                                                    ⅆ              τ                                -                                                    ⅆ                Δ                            ⁢                              xe2x80x83                            ⁢              T                                      ⅆ              τ                                          
respectively, where xcex94T0=T0xe2x88x92TS and xcex94T=Tsxe2x88x92Tr. These results are stored as a function of temperature.
The second experiment uses a pair of calibration samples without pans. The calibration samples may have the same mass, or may have different masses. Preferably, the calibration samples are sapphire samples (e.g., monocrystalline sapphire disks), preferably weighing 25 mg or more. Other reference materials with well-known specific heats and no transitions in the temperature range of the calibration may be used instead of sapphire (in which case Csapph would be replaced in the following equations by Cmat where Cmat is the specific heat of the other reference material).
The sample and reference heat flows from the heat balance equations are set as follows:             q      s        =                  m        s            ·              C        sapph            ·                        ⅆ                      T            ss                                    ⅆ          τ                                q      r        =                  m        r            ·              C        sapph            ·                        ⅆ                      T            rs                                    ⅆ          τ                    
where ms, mr are the masses of the sample and reference sapphires, respectively, Csapph is the specific heat of sapphire and Tss and Trs are the temperatures of the sample and reference sapphire.
Assume:             ⅆ              T        ss                    ⅆ      τ        =                              ⅆ                      T            s                                    ⅆ          τ                    ⁢              xe2x80x83            ⁢      and      ⁢              xe2x80x83            ⁢                        ⅆ                      T            rs                                    ⅆ          τ                      =                  ⅆ                  T          r                            ⅆ        τ            
Substituting for qs and Tss in the sample heat balance equation and solving for the sensor sample thermal capacitance Cs:       C    s    =                    m        s            ·      Csapph                                Δ          ⁢                      xe2x80x83                    ⁢                      T            0                                                (                                          ⅆ                                  T                  s                                                            ⅆ                τ                                      )                    ·                      τ            s                              -      1      
Substituting for qr and Trs in the reference heat balance equation and solving for the sensor reference thermal capacitance Cr:       C    r    =                    m        r            ·              C        sapph                                                  Δ            ⁢                          xe2x80x83                        ⁢                          T              0                                +                      Δ            ⁢                          xe2x80x83                        ⁢            T                                                (                                                            ⅆ                                      T                    s                                                                    ⅆ                  τ                                            -                                                                    ⅆ                    Δ                                    ⁢                                      xe2x80x83                                    ⁢                  T                                                  ⅆ                  τ                                                      )                    ·                      τ            r                              -      1      
The results from the second experiment using sapphire (or another well-known calibration material) using the time constants for DSC cell obtained in the first experiment are then used to calculate the sample and reference sensor heat capacities as a function of temperature. Finally, the sensor sample and reference thermal resistances are calculated from the time constants and the sensor thermal capacitances:       R    s    =                              τ          s                          C          s                    ⁢              xe2x80x83            ⁢      and      ⁢              xe2x80x83            ⁢              R        r              =                  τ        r                    C        r            
A second preferred embodiment is similar to the first embodiment, but uses sapphire (or another material with a well-known heat capacity and no transitions in the temperature range of interest) calibration samples in both the first and the second calibration experiments. The calibration equations and their derivation for this embodiment are described below.
Power Compensation DSCs
As applied to power compensation differential scanning calorimeters, the present invention uses differential temperature measurements, a single temperature measurement, a differential power measurement, and a five term heat flow equation to model the instrument. The present invention is also a method by which the thermal parameters required to apply the five term heat flow equation are determined. Differential scanning calorimeters employing this invention will have empty DSC cell heat flow that is much closer to zero (leading to improved baselines) and will have substantially improved resolution over conventional instruments.
In a preferred embodiment, the two differential temperature measurements are the differential temperature xcex94T0 across thermal resistance Rs, and the differential temperature xcex94T between the sample and reference holders. The absolute temperature of the sample holder and the power difference between the sample and reference holders are also measured (i.e., the differential power to the sample with respect to the reference). Additionally, the four thermal parameters, Rs, Rr, Cs and Cr must be known. The use of two differential temperature measurements allows the use of a heat flow model that includes all five terms of the five term heat flow equation. The heat flow signal that results has improved baseline performance and improved dynamic response. Additionally, because the heat flow signal is greater during a transition, the calorimeter has greater sensitivity.
Other choices of the two differential temperature measurements are also suitable, as explained below.
The present invention also comprises a method by which the four thermal parameters Cs, Cr, Rs, Rr are determined. This determination constitutes heat flow calibration of the DSC.
Heat flow calibration requires two experiments from which the four thermal parameters can be calculated. The first experiment is performed with an empty DSC cell. The DSC program begins with an isothermal temperature segment at a temperature below the lowest temperature of the desired calibration range, followed by a constant heating rate temperature ramp, and ending with an isothermal temperature segment above the highest temperature of the desired calibration range. The heating rate should be the same as the heating rate that is to be used for subsequent experiments. The second calibration experiment is performed with sapphire samples without pans in both the sample and reference holders. The same thermal program is used for the second experiment as was used for the first (empty DSC) experiment. The two calibration experiments and the calculation of the thermal parameters based on the experiments are explained in detail below.
Improved Calculation
The present invention also comprises an improved method for calculating sample heat flow in a differential scanning calorimeter that can be used with both heat flux and power compensation DSCs.
Differential scanning calorimeters employing the improved calculation of the present invention furnish a sample heat flow signal that is a substantially more accurate representation of the sample heat flow during the entire DSC experiment, essentially free of the smearing effects that are present in conventional DSC. Accordingly, DSCs using the present invention will have greatly improved resolution. For example, kinetic analysis requiring partial integration of peak areas can be practiced using the present invention whereas partial integration is of limited use with conventional DSCs, due to the distortions of the sample heat flow signal.
The result is a more accurate measurement of the sample heat flow during transitions in which the heating rate of the sample differs from that of the reference. Resolution is improved because the return to baseline of the heat flow signal at the completion of a transformation is much more rapid.
Modulated DSCs
The ""949 application discloses a DSC in which the sample and reference heat flows may be measured separately. The equations for heat flow to the sample and its pan and to the reference (if used) and its pan are:       q    s    =                              Δ          ⁢                      xe2x80x83                    ⁢                      T            0                                    R          s                    -                                    C            s                    ·                                    ⅆ                              T                s                                                    ⅆ              τ                                      ⁢                  xe2x80x83                ⁢                  q          r                      =                                        Δ            ⁢                          xe2x80x83                        ⁢                          T              0                                +                      Δ            ⁢                          xe2x80x83                        ⁢            T                                    R          r                    -                        C          r                ·                  (                                                    ⅆ                                  T                  s                                                            ⅆ                τ                                      -                                                            ⅆ                  Δ                                ⁢                                  xe2x80x83                                ⁢                T                                            ⅆ                τ                                              )                    
Two temperature differences are used, the conventional xcex94T between sample and reference positions of the sensor and an additional signal xcex94T0 between the sample position and the base of the sensor. Thermal resistances and capacitances, Rs, Rr, Cs and Cr of the sensor are determined form the calibration methods disclosed in that invention. Having the individual heat flow signals allows the sample and reference heat capacities to be measured using the heat capacity equation:       C    sm    =                                          q            s                    _                          ω          ·                                    T              sm                        _                              ⁢              xe2x80x83            ⁢              C        rm              =                            q          r                _                    ω        ·                              T            sm                    _                    
The temperature and heat flow amplitudes are found using the deconvolution algorithm described in U.S. Pat. No. 5,224,775, which is incorporated by reference herein. Measured temperatures in the DSC cell are the temperatures of the sample and reference platforms, not Tsm and Trm as required by the sample and heat capacity equations. However, Tsm and Trm can be found using the equations for heat flow between the pans and the sensor:       q    s    =                                          T            s                    -                      T            sm                                    R          sm                    ⁢              xe2x80x83            ⁢              q        r              =                            T          r                -                  T          rm                            R        rm            
Solving for Tsm and Trm:
Tsm=Tsxe2x88x92qsxc2x7Rsm Trm=Trxe2x88x92qrxc2x7Rrm
Temperatures Ts and Tr are measured, heat flows qs and qr are measured, and predetermined values for Rsm and Rrm are used. Temperature Tr is not measured directly, but is obtained by combining Ts and xcex94T, so that the equation for Trm becomes:
Trm=Tsxe2x88x92xcex94Txe2x88x92qrxc2x7Rrm
It is implied that the temperatures of the sample and its pan are the same and that the temperature of the reference (if used) and its pan are the same. The pan heat capacities are the product of pan masses and the pan material specific heat capacity:
Csp=mspxc2x7cpm Crp=mrpxc2x7cpm
Substitute the sample pan heat capacity into the equation for measured sample heat capacity and solve for the sample heat capacity:
Css=Csmxe2x88x92mpsxc2x7cpm
Substitute the reference pan heat capacity into the equation for measured reference heat capacity and solve for pan heat capacity:       c    pm    =                    C        rm            -              C        rs                    m      pr      
Substitute into the equation for the sample heat capacity:       C    ss    =            C      sm        -                            m          ps                          m          pr                    ·              (                              C            rm                    -                      C            rs                          )            
Finally, substitute the equations for Csm and Crm:       C    ss    =                              q          s                _                    ω        ·                              T            sm                    _                      -                            m          ps                          m          pr                    ·              (                                                            q                r                            _                                      ω              ·                                                T                  rm                                _                                              -                      C            rs                          )            
All quantities on the right hand side are measured as described above, except for the heat capacity of the reference material, which must be known. Most DSC experiments are performed with an empty reference pan. In that case the sample heat capacity equation simplifies to:       C    ss    =                              q          s                _                    ω        ·                              T            sm                    _                      -                            m                      p            ⁢                          xe2x80x83                        ⁢            s                                    m          pr                    ·                                    q            r                    _                          ω          ·                                    T              rm                        _                              